A push to radically reshape the outmoded U.S. air-traffic control system is gaining support, as airlines and some labor unions join to back change and a top lawmaker drafts legislation that could effectively privatize services.

Rep. Bill Shuster, a Pennsylvania Republican and the chairman of the House Transportation and Infrastructure Committee, and his staff are drafting legislation to strip the nation’s 15,000 civilian controllers and more than 230 air-traffic facilities from the Federal Aviation Administration, possibly...